Shape Preserving Fuzzy Interpolation Reasoning Function

Fuzzy reasoning is the core part of fuzzy control theory. At present, the fuzzy reasoning method which being used most is compositional rule of inference (CRI) put forward by professor Zadeh. However, if the rule base is spare, it’s difficult to get the corresponding reasonable inference result. In recent years, some scholars have proposed some new reasoning methods to solve the problem of fuzzy reasoning under the spare rules. But more or less, there are some deficiencies:the reasoning process and the mathematical operation is complicated, some of them can’t guarantee the normality of the consequence.In this paper, the shape preserving fuzzy interpolative reasoning method based on characteristic nodes is not only applicable for the complete rule base, and can also be applied for sparse rule base. First seen the fuzzy rule base as the fuzzy data set composed by the premise and the conclusion, extract the characteristic nodes of the input and output membership functions, get the ordered pairs composed by the characteristic nodes of the premise and the conclusion, then put the ordered pairs as the interpolation nodes, then the interpolative reasoning function is given. As the interpolation curves get by the Lagrange interpolation, Newton interpolation and the other piecewise interpolation methods are general non-shape preserving, The existing shape preserving interpolation algorithms are generally based on monotone sequences, which can guarantee the monotonous, concave and convex, but can’t guarantee the extremum points of the interpolation function are the same as the extremum points of the interpolation nodes. In this paper, based on the existing shape preserving interpolation methods, the a-smooth shape preserving interpolation method for extremum preserving is given for any interpolation sequence. The interpolation results show that the approximation effect with the interpolation nodes, the interpolation function obtained by this method is significantly better than the other shape preserving interpolation methods, in other words, the monotonicity, concavity and convexity, especially the extremum points of them are the same. Choose the shape preserving interpolation method for extremum preserving mentioned in this paper can improve the reasonableness of the interpolative results.For the rule base which the input and output are all triangular membership functions, the specific constructing steps of the shape preserving interpolative reasoning function is given, then the shape preserving interpolation reasoning method based on the characteristic nodes is put forward. At last, in order to verify the feasibility and operability of the fuzzy reasoning algorithm, for the specific example, use the Matlab programming to calculate, the result shows that it is reasonable.The result shows that the reasoning method is simple, which can guarantee normality, the reduction of the characteristic nodes, monotone voice of the consequence and intermediate value, which are the main reasoning requirement of reasonableness. This shows that the reasoning method can be used to the fuzzy inference systems which the rule bases are known, on the other hand, this paper solve the structural problem of the interpolation with shape and extremum preserved in theory.